Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,18)(15,17,19,20)(21,22,27,28,26,23,25,24);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,22)(24,27,28,25);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(22,23)(24,28);(2,4)(21,24,25,23,26,28,27,22)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72320024, #72320186, #72320275 |
| maximal normal subgroup of | #72320024, #72320186, #72320275 |
| maximal quotient of | #72322439 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342381);